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Transcript
Definition of a fluid A fluid is defined as a substance that deforms continuously under the action of a shear stress, however small magnitude present. It means that a fluid deforms under very small shear stress, but a solid may not deform under that magnitude of the shear stress. By contrast a solid deforms when a constant shear stress is applied, but its deformation does not continue with increasing time. In Fig.L1.1, deformation pattern of a solid and a fluid under the action of constant shear force is illustrated. We explain in detail here deformation behaviour of a solid and a fluid under the action of a shear force. In Fig.L1.2, a shear force F is applied to the upper plate to which the solid has F been bonded, a shear stress resulted by the force equals to , where A is the contact A area of the upper plate. We know that in the case of the solid block the deformation is proportional to the shear stress τ provided the elastic limit of the solid material is not exceeded. When a fluid is placed between the plates, the deformation of the fluid element is illustrated in Fig.L1.3. We can observe the fact that the deformation of the fluid element continues to increase as long as the force is applied. The fluid particles in direct contact with the plates move with the same speed of the plates. This can be interpreted that there is no slip at the boundary. This fluid behavior has been verified in numerous experiments with various kinds of fluid and boundary material. In short, a fluid continues in motion under the application of a shear stress and can not sustain any shear stress when at rest. Fluid as a continuum In the definition of the fluid the molecular structure of the fluid was not mentioned. As we know the fluids are composed of molecules in constant motions. For a liquid, molecules are closely spaced compared with that of a gas. In most engineering applications the average or macroscopic effects of a large number of molecules is considered. We thus do not concern about the behavior of individual molecules. The fluid is treated as an infinitely divisible substance, a continuum at which the properties of the fluid are considered as a continuous (smooth) function of the space variables and time. To illustrate the concept of fluid as a continuum consider fluid density as a fluid property at a small region.(Fig.L1.3(a)). Density is defined as mass of the fluid molecules per unit volume. Thus the mean density within the small region C could be equal to mass of fluid molecules per unit volume. When the small region C occupies space which is larger than the cube of molecular spacing, the number of the molecules will remain constant. This is the limiting volume v above which the effect of molecular variations on fluid properties is negligible. A plot of the mean density versus the size of unit volume is illustrated in Fig.L1.3(b). The density of the fluid is defined as lim v v m v L1.1 Note that the limiting volume v is about 109 mm3 for all liquids and for gases at atmospheric temperature. Within the given limiting value, air at the standard condition has approximately 3 107 molecules. It justifies in defining a nearly constant density in a region which is larger than the limiting volume. In conclusion, since most of the engineering problems deal with fluids at a dimension which is larger than the limiting volume, the assumption of fluid as a continuum is valid. For example the fluid density is defined as a function of space (for Cartesian coordinate system, x, y, and z) and time (t ) by x, y, z, t . This simplification helps to use the differential calculus for solving fluid problems. Properties of fluid Some of the basic properties of fluids are discussed belowDensity: As we stated earlier the density of a substance is its mass per unit volume. In fluid mechanic it is expressed in three different ways – (1) Mass density ρ is the mass of the fluid per unit volume (given by Eq.L1.1) Unit- kg / m3 Dimension- ML3 Typical values: water-1000 kg/ m3 Air- 1.23 kg / m3 at standard pressure and temperature (STP) (2) Specific weight, w: - As we express a mass M has a weight W=Mg. The specific weight of the fluid can be defined similarly as its weight per unit volume. w g 3 Unit: N / m Dimension: ML2T 2 L1.2 Typical values; water- 9.810 N / m3 Air- 12.07 N / m3 (STP) (3) Relative density (Specific gravity),S:Specific gravity is the ratio of fluid density (specific weight) to the fluid density (specific weight) of a standard reference fluid. For liquids water at 4 0 C is considered as standard fluid. Sliquid = liquid L1.3 water at 4 C 0 Similarly for gases air at specific temperature and pressure is considered as a standard reference fluid. Sgas = gas gas at STP L1.4 Units: pure number having no units. Dimension:- Mo Lo T o Typical vales : - Mercury- 13.6 Water-1 Specific volume vs : - Specific volume of a fluid is mean volume per unit mass i.e. the reciprocal of mass density. vs 1 Units:- m3 /kg Dimension: M-1L3 Typical values: - Water - 103 m3 /kg Air- 1.23 103 m3 /kg Viscosity In section () definition of a fluid says that under the action of a shear stress a fluid continuously deforms, and the shear strain results with time due to the deformation. Viscosity is a fluid property, which determines the relationship between the fluid strain rate and the applied shear stress. It can be noted that in fluid flows, shear strain rate is considered, not shear strain as commonly used in solid mechanics. Viscosity can be inferred as a quantative measure of a fluid’s resistance to the flow. For example moving an object through air requires very less force compared to water. This means that air has low viscosity than water. Let us consider a fluid element placed between two infinite plates as shown in fig (). The upper plate moves at a constant velocity u under the action of constant shear force F . The shear stress, τ is expressed as lim A 0 F dF A dA L1.5 where, A is the area of contact of the fluid element with the top plate. Under the action of shear force the fluid element is deformed from position ABCD at time t to position AB ’C’D’ at time t t (fig ). The shear strain rate is given by d t 0 t dt Where is the angular deformation. Shear strain rate lim L1.6 From the geometry of the figure, we can define u t For small , tan y Therefore, u t y L1.7 d du L1.8 dt dy The above expression relates shear strain rate to velocity gradient along the y-axis. The limit of both side of the equality gives Newton’s Viscosity Law Sir Isaac Newton conducted many experimental studies on various fluids to determine relationship between shear stress and the shear strain rate. The experimental finding showed that a linear relation between them is applicable for common fluids such as water, oil, and air. The relation is d dt Substituting the relation gives in equation( L1.8 ) du dy Introducing the constant of proportionality du dy L1.9 L1.10 where is called absolute or dynamic viscosity. Dimensions and units for are ML1T 1 and N s / m 2 , respectively. [In the absolute metric system basic unit of coefficient of viscosity is called poise. 1 poise = N s / m 2 ] Typical relationships for common fluids are illustrated in fig*. The fluids that follow the linear relationship given in equation (L1.10) are called Newtonian fluids. Kinematic viscosity v Kinematic viscosity is defined as the ratio of dynamic viscosity to mass density v L1.11 Units: m 2 / s Diminutions: L2T 1 Typical values: water 1.14 106 m2 s 1 air1.46 105 m2 / s Non–Newtonian fluids Fluids in which shear stress is not linearly related to the rate of shear strain are non– Newtonian fluids. Examples are paints, blot, polymeric solution, etc. Instead of the dynamic viscosity apparent viscosity, ap which is the slope of shear stress versus shear strain rate curve, is used for these types of fluid. Based on the behavior of ap , non–Newtonian fluids are broadly classified into the following groups- (a) Pseudo plastics (shear thinning fluids): ap decreases with increasing shear strain rate. For example polymer solutions, colloidal suspensions, latex paints, pseudo plastic. (b) Dilatants (shear thickening fluids) ap increases with increasing shear strain rate. Examples: Suspension of starch and quick sand (mixture of water and sand). (c) Plastics: Fluids that can sustain finite shear stress without any deformation, but once shear stress exceeds the finite stress y , they flow like a fluid. The relation between the shear stress and the resulting shear strain is given by du y ap dy n L1.12 Fluids with n = 1 are called Bingham plastic. some examples are clay suspensions, tooth paste and fly ash. (d) Thixotropic fluid: ap decreases with time under a constant applied shear stress. Example: Ink, crude oils. (e) Rheopectic fluid: ap increases with increasing time. Example: some typical liquid-solid suspensions. Surface tension In this section we will discuss about a fluid property which occurs at the interfaces of a liquid and gas or at the interface of two immiscible liquids. As shown in fig ( ) the liquid molecules- ‘A’ is under the action of molecular attraction between like molecules (cohesion). However the molecule ‘B’ close to the interface is subject to molecular attractions between both like and unlike molecules (adhesion). As a result the cohesive forces cancel for liquid molecule ‘A’. But at the interface of molecule ‘B’ the cohesive forces exceed the adhesive force of the gas. The corresponding net force acts on the interface; the interface is at a state of tension similar to a stretched elastic membrane. As explained, the corresponding net force is referred to as surface tension, . In short it is apparent tensile stresses which acts at the interface of two immiscible fluids. Dimension: MT 2 Unit: N / m Typical values: Water 0.074 N / m at 20 C with air. Note that surface tension decreases with the liquid temperature because intermolecular cohesive forces decreases. At the critical temperature of a fluid surface tension becomes zero; i.e. the boundary between the fluids vanishes. Pressure difference at the interface In order to study the effect of surface tension on the pressure difference across a curved interface, consider a small spherical droplet of a fluid at rest (refer to Fig ##) Since the droplet is small the hydrostatic pressure variations become negligible. The droplet is divided into two halves as shown in Figure##. Since the droplet is at rest, the sum of the forces acting at the interface in any direction will be zero. Note that the only forces acting at the interface are pressure and surface tension. Equilibrium of forces gives P liq Pgas r 2 2 r L1.13 Solving for the pressure difference and then denoting P Pliq Pgas we can rewrite equation (L1.13) as 2 P L1.14 r Contact angle and welting As shown in fig. a liquid contacts a solid surface. The line at which liquid gas and solid meet is called the contact line. At the contact line the net surface tension depending upon all three materials- liquid, gas, and solid is evident in the contact angle, C . A force balance on the contact line yields: gas solid cosc L1.15 where gas is the surface tension of the gas-solid interface, solid is the surface tension of solid-liquid interface, and is the surface tension of liquid-gas interface. Typical values: C 0 for air-water- glass interface C 140 for air-mercury–glass interface If the contact angle c 90 the liquid is said to wet the solid. Otherwise, the solid surface is not wetted by the liquid, when c 90 . Capillarity If a thin tube, open at the both ends, is inserted vertically in to a liquid, which wets the tube, the liquid will rise in the tube (fig). If the liquid does not wet the tube it will be depressed below the level of free surface outside. Such a phenomenon of rise or fall of the liquid surface relative to the adjacent level of the fluid is called capillarity. If c is the angle of contact between liquid and solid, d is the tube diameter, we can determine the capillary rise or depression, h by equating force balance in the z-direction (shown in Fig#), taking into account surface tension, gravity and pressure. Since the column of fluid is at rest, the sum of all of forces acting on the fluid column is zero. The pressure acting on the top curved interface in the tube is atmospheric, the pressure acting on the bottom of the liquid column is at atmospheric pressure because the lines of constant pressure in a liquid at rest are horizontal and the tube is open. Upward force due to surface tension cos c d d h 4 Thus equating these two forces we find Weight of the liquid column g. L1.16 2 d2 cos c d g h 4 The expression for h becomes 4 cos c h gd L1.17 L1.18 L1.19 Typical values of capillary rise are (a) Capillary rise is approximately 4.5 mm for water in a glass tube of 5 mm diameter. (b) Capillary depression is approximately –1.5 mm (depression) for mercury in the same tube. Capillary action causes a serious source of error in reading the levels of the liquid in small pressure measuring tubes. Therefore the diameter of the measuring tubes should be large enough so that errors due to the capillary rise should be very less. Besides this, capillary action causes the movement of liquids to penetrate cracks even when there is no significant pressure difference acting to move the fluids in to the cracks. In figure (), a two-dimensional model for the capillary rise of a liquid in a crack width ‘b’ is illustrated. The height of the capillary rise can also be computed by equating force balance as explained in the previous section. h 2 cos c b g L1.20 Vapour Pressure Since the molecules of a liquid are in constant motion, some of the molecules in the surface layer having sufficient energy will escape from the liquid surface, and then changes from liquid state to gas state. If the space above the liquid is confined and the number of the molecules of the liquid striking the liquid surface and condensing is equal to the number of liquid molecules at any time interval becomes equal, an equilibrium exists. These molecules exerts of partial pressure on the liquid surface known as vapour pressure of the liquid, because degree of molecular activity increases with increasing temperature. The vapour pressure increases with temperature. Boiling occurs when the pressure above a liquid becomes equal to or less then the vapour pressure of the liquid. It means that boiling of water may occur at room temperature if the pressure is reduced sufficiently. For example water will boil at 60 C temperature if the pressure is reduced to 0.2 atm. Cavitation In many fluid problems, areas of low pressure can occur locally. If the pressure in such areas is equal to or less then the vapour pressure, the liquid evaporates and forms a cloud of vapour bubbles. This phenomenon is called cavitation. This cloud of vapour bubbles is swept in to an area of high pressure zone by the flowing liquid. Under the high pressure the bubbles collapses. If this phenomenon occurs in contact with a solid surface, the high pressure developed by collapsing bubbles can erode the material from the solid surface and small cavities may be formed on the surface. The cavitation affects the performance of hydraulic machines such as pumps, turbines and propellers. Compressibility and the bulk modulus of elastics When a fluid is subjected to a pressure increase the volume of the fluid decreases. The relationship between the change of pressure and volume is linear for many fluids. This relationship may be defined by a proportionality constant called bulk modulus. Consider a fluid occupying a volume V in the piston and cylinder arrangement shown in figure. If the pressure on the fluid increase from p to p p due to the piston movement as a result the volume is decreased by V . We can express the bulk modulus of elasticity k p v/v L1.21 The negative sign indicates the volume decreases as pressure increases. As in the limit as p 0 then k Since dp dv / v L1.22 dv dp the equation can be rearranged as v p k dp d / Dimension :- ML1T 2 Unit :- N / m2 Typical values:Air - 1.03 x 105 N/m2 water 2.05 109 N / m2 at standard temperature and pressure as compared to that of Mild steel 2.06 1011 N / m2 . The above typical values show that the air is about 20,000 times more compressible than water while water is about 100 times more compressible than mild steel. Basic Equations To analysis of any fluid problem, the knowledge of the basic laws governing the fluid flows is required. The basic laws, applicable to any fluid flow, are: (a) Conservation of mass. (Continuity) (b) Linear momentum. (Newton’s second law of motion) (c) Conservation of energy (First law of Thermodynamics) Besides these governing equations, we need the state relations like P, T and appropriate boundary conditions at solid surface, interfaces, inlets and exits. Note that all basic laws are not always required to any one problem. These basic laws, as similar in solid mechanics and thermodynamics, are to be reformulated in suitable forms so that they can be easily applied to solve wide variety of fluid problems. System and control volume A system refers to a fixed, identifiable quantity of mass which is separated from its surrounding by its boundaries. The boundary surface may vary with time however no mass crosses the system boundary. In fluid mechanics an infinitesimal lump of fluid is considered as a system and is referred as a fluid element or a particle. Since a fluid particle has larger dimension than the limiting volume (refer to section fluid as a continuum). The continuum concept for the flow analysis is valid. A control volume is a fixed, identifiable region in space through which fluid flows. The boundary of the control volume is called control surface. The fluid mass in a control volume may vary with time. The shape and size of the control volume may be arbitrary.